A view through the walls of our classroom. This is an interactive learning ecology for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.

Subscribe

Powered By

Saturday, February 7, 2009

SINE! COSINE! TANGENT! INVERSE?!? ... OR An Introduction to the Inverses of the Trigonometric Functions

OUTLINE:

Definitions of many-to-one and one-to-one functions

Unit circle

Sine, Cap-Sine, Arcsin!

Cosine, Cap-Cosine, Arccos!

Tangent, Cap-Tangent, Arctan!

Using Right Triangles

SLIDE 2: BLAST FROM THE PAST! GRADE 12 UNIT CIRCLE UNIT!

Review of grade 12 precalculus (unit circle unit)

Ask yourself what angle you're looking for and imagine the unit circle

We gave the answers to the above questions, but Mr.K then says that one of the solutions in each answer is wrong, but why?

Introducing the inverses of the trigonometric functions!

SLIDE 3: VOILA! THE UNIT CIRCLE!

SLIDE 4: SINE! CAP SINE! ARCSIN!

We have the sine function. We want to know its inverse, but we can't since it's a many-to-one function.

If we restrict the domain of the sine function to [-pi/2, pi/2], we have a function that has the same range as the sine function. This restricted sine function is the Sine (pronounced "Cap-Sine") function.

Take the inverse of Sine (switch the x- and y-coordinates). We get the arcsine function.

Note that Sine takes up Quadrants I and IV of sine. Thus, only values of sine found in QI and QIV are in the domain of Sine.

SLIDE 5: COSINE! CAP COSINE! ARCCOS!

We have the cosine function. We want to know its inverse, but we can't since it's a many-to-one function.

If we restrict the domain of the cosine function to [0, pi], we have a function that has the same range as the cosine function. This restricted cosine function is the Cosine (pronounced "Cap-Cosine") function.

Take the inverse of cosine (switch the x- and y-coordinates). We get the arcsine function.

Note that Cosine takes up Quadrants I and II of cosine. Thus, only values of cosine found in QI and QII are in the domain of Cosine.

SLIDE 6: TANGENT! CAP TAN! ARCTAN!

We have the tangent function. We want to know its inverse, but we can't since it's a many-to-one function.

If we restrict the domain of the tangent function to [-pi/2, pi/2], we have a function that has the same range as the tangent function. This restricted cosine function is the Tangent (pronounced "Cap-Tangent") function.

Take the inverse of tangent (switch the x- and y-coordinates). We get the arctan function.

Note that Cosine takes up Quadrants I and IV of tangent. Thus, only values of tangent found in QI and QIV are in the domain of Tangent.

SLIDE 7 TO 12: A BARRAGE OF QUESTIONS!

We found the values of each using the unit circle. Note that some values of the unit circle aren't in the domain of the "Cap functions."

3a and 3b are undefined because the angle doesn't live in the domain of Sine and Cos.

Sine and sine, Cosine and cosine, and Tangent and tangent are inverses of each other. Thus, they undo each other, like multiplying a number by 3 then dividing the number by 3.

SLIDE 13: USING RIGHT TRIANGLES!

In 8a, we let sin^-1 2/3 = x to make a simplified expression cot x. Looks easier to solve, right? We used the definition of sine (the ratio of opposite side to the hypotenuse side) to determine what cot x is. We solved for cot x.

HOUSEKEEPING:

Next scribe is Kristina.

And don't forget to listen to the Dr. Love messages during the last ten minutes of class!